If your interests in music are similar to mine, this is how you can find and enjoy new music; it's free and absolutely legal.

Amazon.com
Amazon.com offers free MP3 downloads, but the list of songs can be overwhelming. You can use the navigation panel to narrow songs by genre:

In addition to individual songs, Amazon.com also offers free albums. To find free albums, use the Sort by Price: Low to High option.

Amazon normally offers high quality MP3 files (above 192 Kbps).

Podcasts
A number of radio stations offer one-song-a-day podcast downloads, which are normally of lower quality (128 Kbps - 192 Kbps), but they should sound fine to most ears. Sure most of the downloads are crap, but occasionally, you may stumble upon a gem. There must be more podcasts available, but here are the ones from NPR that I like:

To download podcasts, you can use a device-specific program, such as iTunes or Zune, but I prefer a free, open-source program called Juice.

Once you find a track that you like among the downloads, you can use an application like Mp3Tag to edit ID3 tags (you may also want to rename file names, because the downloaded files normally have cryptic names).

Use RSS tools
You can automate the process of searching for free MP3 downloads across multiples sources using RSS tools, such as Yahoo! Pipes. The key is to subscribe to the right aggregation service. I would recommend the SlickDeals' Freebes and Hot Deals forums, as well as Front Page (each of these has a subscribe/RSS link). You can follow this tutorial (and this one) to set up a Yahoo! Pipe filtering only posts containing music-specific keywords, such as free, MP3, music, album, etc.

Keep your eyes (and ears) open
Many vendors offer free promotional MP3s and CDs, some of which are not too bad. For example, I do not remember how I got it, but somehow I received a free Mommy Mix CD from Safeway.

Being a promotional CD, I did not expect much from it, but it it ended up being one of my family's favorite music CDs (in fact, if it were available for sale, I would've bought it). So don't assume that promotional music is always crappy, some may surprise you.